KINEMATICS. [ 



94 



The period is therefore independent of the amplitude a. Il 

 follows that two simple harmonic motions resulting from tw< 

 uniform circular motions of the same angular velocity on two I 

 concentric circles of different radii have the same period ; such] 

 motions are called isochronous. 



174. If the time / be counted, not from A, but from some 

 other point P Q on the circle for which %AOP Q = e, we have 

 ^AOP=(0t+e, and the equation of the simple harmonic 



motion is 



05) 



The angle o>/+e is called the phase-angle, while e is the epoch-j 

 angle, of the motion. The names phase and epoch are sometimesj 

 applied to these angles, although, strictly speaking, the phase is 

 the time (usually expressed as a fraction of the period T) oi\ 

 passing from the position A of maximum displacement to any 

 position P* while the epoch is the phase corresponding to the 

 time t=o. 



175. Differentiating equation (15), we find the velocity 



v x = = -aco sin (orf+e) ; (i6)l 



at 



and differentiating again, we obtain the acceleration 



(I?) 



of simple harmonic motion. 



The same values can be derived by projecting the velocity 

 and acceleration of the uniform circular motion of P on the 

 diameter A A ', as is readily seen- from Fig. 44. 



176. Equation (17) shows that in simple harmonic motion the 

 acceleration is directly proportional to the distance from the 

 centre. 



